Significant Figures & Uncertainty

Every instrument reports in steps. A metre rule marked in millimetres, a caliper stepping in 0.01 mm, a timer ticking in 0.001 s: a rounded digital indication locates its unrounded value within one display step, while analogue scales may permit interpolation. When a single reading is all you have, that step defines an interval of half-width a, and if the value is equally likely anywhere inside it the standard uncertainty is a/√3 — about 0.29 of a step. This is the resolution-only contribution under that rounding model; calibration and other effects may add uncertainty. It says nothing about whether the instrument is telling the truth: a caliper displaying 0.01 mm steps while sitting 0.15 mm off zero has fine resolution but a biased indication.

Time the same pendulum ten times and you get ten different numbers. The best estimate is the mean x̄. The spread is the sample standard deviation s, with N − 1 in the denominator because x̄ was taken from the same data and one degree of freedom is already spent. s answers a question about a single reading — how far does one timing wander? — and it does not shrink as data accumulates; it only becomes better determined. For independent, stable readings, the estimated random standard uncertainty of the mean is u(x̄) = s/√N. Calibration contributions are assessed separately. Four times the readings halve it, nine times reduce it to a third. That √N is also why brute repetition stops paying: going from 100 timings to 400 buys one more factor of two.

A measurement is a claim about an interval, not a point. Every instrument has a smallest step it can resolve, and repeating a reading shows how far the result moves when nothing about the object has changed. Random effects scatter readings around the measurement-process mean, which may itself be biased and give a random contribution to the mean that falls as 1/√N for independent, stable readings; systematic effects shift every reading the same way and survive any number of repeats. So a result is finished only when it is written as a best estimate, an uncertainty in the same unit, and a stated coverage: x = x̄ ± u. Derived quantities inherit those intervals through the partial derivatives of the formula — absolute uncertainties combine for sums, relative ones for products and powers. Significant figures are only the shorthand: once u is known, round u first, then quote the value to the same decimal place. The rest of what the calculator prints is decoration.

Precision is agreement among your own readings; accuracy is agreement with the true value. Both scatter and bias matter: precise readings can still be far from the reference value. Random effects — reaction time, draughts, a last digit flickering — push readings both ways, so they appear as scatter and give a random contribution to the mean that falls as 1/√N for independent, stable readings. Systematic effects — a stretched tape, an uncorrected zero, a clock running 0.5% fast, a scale read at an angle — shift every reading by nearly the same amount. Repetition cannot find them, because they sit inside x̄ and barely touch s. They are exposed only by changing something: calibrate against a standard, measure the quantity a second way, reverse the instrument end for end, or check the zero before and after the run.

Density comes from a mass and two lengths; a spring constant from a force and an extension. If f depends on independent inputs x, y, …, its uncertainty follows from the partial derivatives: u_f² = (∂f/∂x)²uₓ² + (∂f/∂y)²u_y² + … . Each partial derivative measures how hard f leans on that input, and the squares mean the terms add in quadrature rather than linearly, because uncorrelated contributions have zero covariance in this first-order variance model. Two shortcuts follow, and they cover most of this course. For sums and differences, absolute uncertainties combine in quadrature. For products, quotients and powers, relative uncertainties do, with each exponent multiplying its own term — so compare the weighted contributions before deciding which measurement to improve.

Take a cylinder: m = 44.75 ± 0.02 g, d = 12.70 ± 0.02 mm, L = 50.00 ± 0.05 mm. With V = πd²L/4 the calculator gives ρ = m/V = 7065.2 kg m⁻³. The relative terms are 2(0.02/12.70) = 0.315% for the diameter, 0.05/50.00 = 0.100% for the length and 0.02/44.75 = 0.045% for the mass; in quadrature they give 0.333%, so u_ρ = 0.00333 × 7065.2 ≈ 24 kg m⁻³. The diameter, entering squared, carries almost all of it; the first improvement should target diameter measurement, subject to checking correlations and other effects. Now report. Round u to one significant figure, or two when one would discard too much — here 24 kg m⁻³ — then round the value to that same decimal place: ρ = 7065 ± 24 kg m⁻³, or (7.065 ± 0.024) × 10³ kg m⁻³.

• Keep guard digits throughout the calculation; round the final uncertainty and value together.
• Leading zeroes are not significant: 0.00450 has three significant figures.
• Trailing zeroes can convey precision: 1.200 has four significant figures; scientific notation clarifies ambiguous integers.
• When only digit-based rules are available: multiplication/division follows significant figures; addition/subtraction follows decimal places. They do not replace a proper uncertainty budget.
• Defined constants and exact counts do not impose a significant-figure limit.

Significant figures are shorthand for an uncertainty, not a replacement for one. Copying 7065.2204 kg m⁻³ off the calculator does not make the density good to 0.0001 kg m⁻³, and trimming it to 7.07 × 10³ kg m⁻³ still leaves the reader guessing whether the interval is ±5 or ±50. Digit counting — first non-zero digit to the last one claimed, so 7065 asserts four and 7.07 × 10³ asserts three — keeps arithmetic tidy. When the number carries a claim someone else will build on, propagate the uncertainties and let u fix the last digit.